Theorems on the Fredholm alternative and well-posedness of the linear boundary value problem $u'(t)=\ell(u)(t)+q(t)$ , $h(u)=c$ , where $\ell:C([a,b];\mathbb{R})\rightarrow L([a,b];\mathbb{R})$ and $h:C([a,b];\mathbb{R}) \rightarrow \mathbb{R}$ are linear bounded operators, $q\in L([a,b]; \mathbb{R})$ , and $c\in \mathbb{R}$ , are established even in the case when $\ell$ is not a strongly bounded operator. The question on the dimension of the solution space of the homogeneous equation $u'(t)=\ell(u)(t)$ is discussed as well.